This allows to compute depth and spectral bottom reflectance of shallow pixels.

Using a value Ki/j observed in the image, operational two-way spectral attenuation coefficients are interpolated from Jerlov's table of diffuse attenuation coefficients for downwelling irradiance: this allows to compute depths in meters.

All this can be done ahead of any field work.

For a proper "finish", all computed depths then need to be multiplied by a final adjustment factor which can only be derived from some sea truth when available, while spectral bottom reflectances remain unaffected.

The 4SM process is a "ratio method" derived from the concepts behind "passive multispectral bathymetry modeling"

Modeling refers to the estimation of the desired information using simplifying assumptions in order to operate the very complex physical model of radiative transfer of light through water for an applied purpose.

Although modeling may not be presented in simple plain terms,

it is necessary to present its nature and detail its limitations in respect of the services offered.

LB is the BOA water column corrected radiance for the shallow bottom (i.e. if Z=0)

LsB=LB+La is the TOA water column corrected radiance for the shallow bottom (i.e. if Z=0)

Please note

Physical units of radiance

Radiance terms in this model do not need to be specified in physical units of radiance. Radiances may be raw digital numbers.

Atmospheric correction to BOA reflectance

If K, La and Lsw are assumed to be constant over the remote sensing scene and may be estimated from the image, then the image does not need to be corrected to BOA reflectance for atmospheric path radiance

Ratio method

Using a band ratio method to derive both Z and spectral LB only requires that spectral LB meets a certain condition.

Like most existing operational propositions,

the 4SM process assumes that the deep water radiance Lsw may be estimated from the imagery itself.

Unlike all current methods,

the calibration of the optical properties of the model in 4SM

accounts for the color of the water column,

and does not need or use any field data.

In an innovative approach in 4SM:

SL: the concept of the "soil line"

uses the bare land areas of the image to derive an average spectral model of the desired water column corrected image: a spectral bottom reflectance reference model at null depth.

This shall be used to specify the above mentioned required condition on spectral LB,

which, among other things, eliminates the need for a spectral library of bottom type end-members to be collected on site.

This is a distinct improvement of Polcyn et al.'s work (1970).

La: estimating the atmospheric path radiance

The soil line also allows for the estimation of the spectral atmospheric path radiance La,

and therefore of the spectral water volume backscattering radiance Lwin Lsw=La+Lw,

thus permitting a first order atmospheric correction of the imagery at the base of the atmosphere.

Using one of these ratios and Jerlov's published data on marine optics (1976),

and in line with Kirk's statement that "a family of curves, of progressively changing shape, determined mainly by phytoplancton concentration, is observed. Thus, for any given oceanic water, specification of the ratio of radiances or radiance reflectances at any two wavelengths should, in effect, specify the whole radiance reflectance curve, and therefore the optical character of the water ",

one then derives one seed K value, either Ki or Kj ,

which then is enough for specifying all required spectral K values across the whole visible range (400-700 nm).

This avoids the need for field calibration data,

and is enough to "ballpark" the optical calibration very closely.

Interestingly, the estimated spectral K values are very close to sea-truth derived values.

To this extent, it means that 4SM is NOT site-specific

Still, the spectral bottom reflectances only depend on the series of ratios

as seed value only needs to be realistic for computing shallow water depth in meters

Now that all model parameters are specified

Z and spectral LB are then derived at each shallow pixel in the image by inversion of the above model , assuming that, for the assumed depth Z, the average spectral water column corrected bottom signature LB must approximately match the spectral soil line.

The output are

a DTM of the shallow water area,

and also " a low-tide view of the scene in units of radiance" ready for shallow bottom typing by current thematic classification methods.

Quite close: interestingly, the estimated spectral K values are very close to sea-truth derived values.

Some sea-truth data is needed though for fine-tuning the estimation of Z,